Problem 186 - Elementary Algebra

Question

In the following exercises, translate to a system of equations and solve the system. The sum of two numbers is negative sixteen. One number is seven times the other. Find the numbers.

Step-by-Step Solution

Answer

The numbers are -14 and -2.

Step 1: Define the variables

Let the first number be denoted by x and the second number by y.

Step 2: Set up the first equation

The problem states that the sum of the two numbers is negative sixteen. This can be written as: \[ x + y = -16 \]

Step 3: Set up the second equation

The problem also states that one number is seven times the other. This can be written as: \[ x = 7y \]

Step 4: Substitute and solve

Substitute the expression for x from the second equation into the first equation: \[ 7y + y = -16 \] Combine like terms: \[ 8y = -16 \] Solve for y: \[ y = -2 \]

Step 5: Find the value of x

Use the value of y to find x using the second equation: \[ x = 7y \] Substituting y = -2: \[ x = 7(-2) \] \[ x = -14 \]

Key Concepts

Translation of Word Problems

When given a word problem, understanding the language and forming equations from the descriptions is crucial. In this case, the problem described a relationship between two numbers and a specific sum, requiring us to translate this into mathematical terms. A good approach is to identify keywords:

* **Sum** implies addition.
* **Times** implies multiplication.

Identifying the variables (numbers in question) and expressing them using these keywords helped set up the system of equations. The sentence _'The sum of two numbers is negative sixteen'_ translates to the equation \( x + y = -16 \). Similarly, _'one number is seven times the other'_ translates to \( x = 7y \). Always strive to break down the sentences carefully to correctly translate word problems into equations.

Linear Equations

Linear equations are pivotal in solving systems of equations, especially when translating word problems. A linear equation is an algebraic equation where each term is either a constant or the product of a constant and a single variable. The equations we encountered are both examples of linear equations:

* The first equation is: \( x + y = -16 \)
* The second equation is: \( x = 7y \).

These linear equations form a system because they contain the same variables, x and y. The ultimate goal is to find values for these variables that satisfy both equations simultaneously. Understanding the properties of linear equations and practicing how to manipulate them through addition, subtraction, and substitution is key.

Substitution Method

The substitution method is a technique to solve systems of linear equations. It involves substituting one equation into another, making it easier to solve for one variable. In our problem, we start with the equations:

* \( x + y = -16 \)
* \( x = 7y \)

Using the second equation, which is already solved for x, we substitute \( 7y \) for x in the first equation. This substitution results in:

\[ 7y + y = -16 \]

This simplifies to:
\[ 8y = -16 \]

Now, the equation only has one variable, y, which can be solved easily. The substitution method is essential for efficiently solving systems where one equation is already isolated for a variable.

Solving for Variables

Once we have an equation with a single variable, solving it becomes straightforward. From our substitution step, we derived:

\[ 8y = -16 \]

To find y, divide both sides by 8:
\[ y = -2 \]

With y known, we can substitute its value back into one of the original equations to find x. Using \( x = 7y \):

Substituting \( y = -2 \):
\[ x = 7(-2) \]
\[ x = -14 \]

Therefore, the values of the numbers are \( x = -14 \) and \( y = -2 \). Breaking down each step and isolating variables is a fundamental skill for solving any system of equations efficiently.